In this chapter, we again have ω as the set of agents, but we only consider one-way visibility on ω (in which each agent can see some subset of the higher-numbered agents). We first use the μ-predictor from Chapter 1 to completely characterize those transitive graphs adequate for minimal and optimal predictors. The result for optimal predictors is then extended to nontransitive graphs (in the countable case).
The case of minimal predictors for nontransitive graphs turns out to be quite complex. We explore a particular nontransitive visibility graph called the “parity relation” in which even-numbered agents (in ω) see higher-numbered odds and vice versa. It turns out here that whether of not there is a predictor ensuring at least one correct guess is very dependent on the number of colors: with finitely many colors, the answer is yes, with ℵ2 colors, the answer is no, and with denumerably many colors the answer is independent of ZFC plus a fixed value of the continuum.
Section 4.6 deals with P-point ultrafilters and Ramsey ultrafilters on ω. Section 4.7 provides a finer analysis of the investigations begun in Section 4.6. The chapter concludes with a brief glimpse at the work done by Andreas Blass and others on evasion and prediction.
Blass, A.: Reductions between cardinal characteristics of the continuum. In: Bartoszyński, T., Scheepers, M. (eds.) Contemporary Mathematics 192: Set Theory: Annual Boise Extravaganza in Set Theory (BEST) Conference, pp. 31–49 (1996)Google Scholar